1
Population is part of the outcome
An outcome now lists a number of people as well as their welfare. Total welfare rewards adding people whose lives are worth living; average welfare is unmoved by numbers and moved only by the level.
Lesson · Ethics and the Arithmetic of Welfare · Consequentialism
Total, Average and the Repugnant Conclusion
When outcomes contain different numbers of people, total and average welfare come apart. Compute the smallest population at a barely-worth-living welfare level that beats a flourishing one by total, state the mere addition premise behind it, and see what averaging costs instead.
ε: the welfare of a life barely worth living
n·ε adds up: enough people beat any total
mere addition: add lives, harm no one
average: one person at 11 beats a crowd at 10
Population and the sum
Core ideas
What this lesson establishes
Once the number of people can vary, “maximise welfare” stops being one instruction. Each way of making it precise has a consequence the others avoid.
2
The repugnant conclusion
For any flourishing population and any positive welfare level, however small, some larger population at that level has the bigger total. The lab computes the smallest such population.
3
Mere addition is a premise
Adding people whose lives are worth living, at no cost to anyone else, does not make an outcome worse. The total accepts it. The average does not.
The material
Adding people
A flourishing population, a crowd at the margin, and two rules that part company.
Take a population of three people at welfare 10 each: A = 10, 10, 10. Its total is 30 and its average is 10. Now consider a population whose members all live at welfare ε, a small positive number standing for a life barely worth living. Let ε = 1/10.
A crowd of n people at 1/10 has a total of n·(1/10). That exceeds 30 once n is large enough, and the lab finds the least such n.
n = 300 total 30 ties with A
n = 301 total 301/10 exceeds 30
Definition · The repugnant conclusion
For every population at welfare 10 there is a population at a lower positive welfare ε whose total is greater. The least size is the smallest whole number n with n·ε greater than the total; the lab prints it as n*. Here n* = 301.
The lab lists at most twelve people on a side, so B holds twelve of the crowd as a sample and the last tile reports how many would be needed. Twelve at 1/10 total 6/5 and lose to the three at ten; it is the 301 that win. Under total welfare the 301 are better than the three, though every one of them has a life of little value. That is the conclusion Parfit called repugnant.
It follows from a premise that sounds harmless.
Definition · Mere addition
Adding people whose lives are worth living, without making anyone else worse off, does not make an outcome worse.
In the mere-addition preset, B is A plus two people at welfare 2. Total welfare ranks B above A, 34 against 30, and agrees with the premise. Average welfare ranks A above B, since 10 exceeds 34/5, so it denies the premise: two new people with lives worth living made the outcome worse.
That is how averaging dodges the conclusion, and it is not free. In the average-trap preset, A is one person at 11 and B is twelve people at 10. Total welfare prefers B, 120 to 11. Average welfare prefers A, 11 to 10, and the same holds for a million people at 10, which the lab is too small to list. One person at 11 is better than any crowd at 10.
There is no free exit here. Accept the total’s conclusion, accept the average’s, or look for a rule that avoids both. The next lessons tune the weighting; none of them is announced as the answer.
The scale carries a stipulation. “Barely worth living” means a welfare level just above zero, and the computation depends on where zero lies. The lab takes ε as given and cannot tell you whether a life at ε is worth living.
Interactive
Two distributions, six rules
the same people ranked by total, average, priority, the worst off, and sufficiency
Method
Comparing populations of different sizes
Five steps; the second shows where total and average disagree.
1
Write both lists, with their sizes
Count the people as well as listing them. A comparison between populations starts from different lengths.
2
Compute total and average
The total is the sum; the average is the sum divided by the count. Compare the outcomes under each rule and see whether they agree.
3
Name the premise the disagreement turns on
If total ranks the larger population higher and average the smaller, ask whether adding the extra people made things better or worse, and whether they harmed anyone.
4
Choose ε and compute n*
Take the welfare of a barely worthwhile life and find the least number of people at that level whose total exceeds the flourishing population’s.
5
Count the cost
Each rule rejects a premise. Say which premise you would give up and what you would then have to accept.
Total, average and the crowd
Score 0 / 0Choose the answer you can justify.
The smallest crowd that beats three at ten
Fix ε at one tenth and find the least population whose total exceeds 30.
A = 10, 10, 10 total 30 average 10
ε = 1/10 a life barely worth living
n people at ε total n times 1/10
n = 300 total 30 ties with A
n = 301 total 301/10 beats A
so n* = 301
Average welfare, meanwhile, ranks the 301 far below the three: 1/10 against 10. Which of those two verdicts to keep is the question the two rules are answers to.
Common mistakes
1
Assuming averaging avoids the problemAverage welfare avoids the crowd at ε, and then ranks one person at 11 above twelve at 10, and ranks two worthwhile added lives as a loss. It trades one repugnant verdict for others.
2
Reading the conclusion as requiring miseryThe people at ε have lives that are worth living, if barely. If ε were zero or negative no population would be large enough, and the lab prints a dash. The conclusion is about many small goods, not about suffering.
3
Treating n* as a predictionThe 301 depends on the scale and on the choice of ε. Change either and the number moves. It measures what the rule commits you to, not how many people the world will contain.
Check your understanding
✓
Finish when you can compute n* and name the rule that blocks it.
Given a flourishing population and a welfare level ε, you should be able to compute the least population at ε that beats it by total, state the mere addition premise, and say which rule gives it up and what that rule then ranks.
Note: The repugnant conclusion has many proposed escapes, including critical-level views and rules that count a person’s welfare relative to how many exist. The lab compares total and average because they show the structure most clearly.